Parameter estimation in linear regression driven by a Wiener sheet

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1 Ales Mthemtice et Iformtice ) pp. 3 5 Proceedigs of the Coferece o Stochstic Models d their Applictios Fculty of Iformtics, Uiversity of Debrece, Debrece, Hugry, August 22 24, 20 Prmeter estimtio i lier regressio drive by Wieer sheet Sádor Br, Kig Sikoly Fculty of Iformtics, Uiversity of Debrece Kssi út 26, H-4028 Debrece, Hugry br.sdor@if.uideb.hu, sikoly.kig@if.uideb.hu Dedicted to Mátyás Artó o his eightieth birthdy Abstrct The problem of estimtig the prmeters of lier regressio Zs, t) = m g s, t) m pg ps, t) W s, t) bsed o observtios of Z o sptil domi G of specil shpe is cosidered, where the drivig process W is stdrd Wieer sheet d g,..., g p re kow fuctios. We provide expressio for the mximum likelihood estimtor of the ukow prmeters bsed o the observtio of the process Z o the set G. Simultio results re lso preseted, where the drivig rdom sheets re simulted with the help of their Krhue-Loève expsios. Keywords: Wieer sheet, mximum likelihood estimtio, Rdo-Nikodym derivtive. MSC: 60G60; 62M0; 62M30.. Itroductio The Wieer sheet is oe of the most importt exmples of Gussi rdom fields. It hs vrious pplictios i sttisticl modellig. Wieer sheet ppers s limitig process of some rdom fields defied o the iterfce of the Isig model [2], it is used to model rdom polymers [9], to describe the dymics of Heth Jrrow Morto type forwrd iterest rte models [0] or to model rdom mortlity surfces [6]. Further, [7] cosiders the problem of estimtio of the me Reserch hs bee supported by the Hugri Scietific Reserch Fud uder Grt No. OTKA T07928/2009 d prtilly supported by TÁMOP 4.2../B-09//KONV /IK/IT project. The project is implemeted through the New Hugry Developmet Pl co-ficed by the Europe Socil Fud, d the Europe Regiol Developmet Fud. 3

2 4 S. Br, K. Sikoly i oprmetric regressio o two-dimesiol regulr grid of desig poits d costructs Wieer sheet process o the uit squre with drift tht is lmost the me fuctio i the oprmetric regressio. I this pper we cosider lier regressio drive by Wieer sheet, tht is rdom field Zs, t) := m g s, t) m p g p s, t) W s, t).) observed o domi G, where g,..., g p re kow fuctios d W is stdrd Wieer sheet, d we determie the mximum likelihood estimtor MLE) of the ukow prmeters m,..., m p. I priciple, the Rdo-Nikodym derivtive of Gussi mesures might be derived from the geerl Feldm-Hjek theorem [], but i most of the cses explicit clcultios cot be crried out. For exmple, if p = d g shifted Wieer sheet), the MLE of the ukow prmeter is give i [3] d the estimtor is expressed s fuctio of usully ukow rdom vrible stisfyig some chrcterizig equtio. I severl cses the exct form of this rdom vrible c be derived by method proposed i [4] bsed o lier stochstic prtil differetil equtios. Specil cse : Br et l. [3] studied the cse, whe p = d the rdom field Z is observed o rectgulr domi G := [, 2 ] [b, b 2 ], [, 2 ], [b, b 2 ] 0, ). Assumig tht g is bsolutely cotiuous with respect to the Lebesgue mesure d 2 g L 2 G), they proved tht the MLE of the shift prmeter m hs the form m = A ζ, where A = g2, b ) b G 2 [ g u, b )] 2 b du [ 2 g u, v)] 2 dudv, ζ = g, b )Z, b ) b G 2 2 g u, v)zdu, dv), b 2 b [ 2 g, v)] 2 g u, b ) b Zdu, b ) dv.2) b 2 b 2 g, v) Z, dv).3) d it hs orml distributio with me m d vrice /A. For g we hve m = Z, b ). Specil cse 2: Artó N.M. [2] cosidered the cse p = d g d usig Rozov s method foud the MLE of the shift prmeter m whe the process is observed o specil domi G G := {s, t) R 2 : s b, t γs) or s > b, t γb)}

3 Prmeter estimtio i lier regressio drive by Wieer sheet 5 cotiig ε strip of Γ := {s, γs)) : s, b)}, i.e. for some ε > 0 {s, t) R 2 : s [, ε], t [γs), γ)] or s [ ε, b], t [γs), γs) ε]} G, where γ : [, b] 0, ) R is cotiuous d strictly decresig with γb) > 0. Br et l. [4] cosidered the sme model d uder much weker coditios γ C 2, b), γ ) := lim s γ s) [, 0] d γ b) := lim s b γ s) [, 0] exist, d b γ s)γ s) γ s) 2 ds <, ) 2 they proved the result of [2, Theorem 2]. They showed tht the MLE of the shift prmeter m hs the form m = A ζ, where A = b bγb) ds s 2 γs), ζ = c Z, γ)) c 2 Zb, γb)) Γ y Z Γ y 2 Z,.4) c, c 2 re costts depedig o γ d γ t d b, y d y 2 re fuctios of γ, γ, γ, d Z deotes the orml derivtive of Z [4, Defiitio 4.]. If γ ) = we hve ζ = Zb, γb)) bγb) b Zs, γs)) b s 2 ds γs) Zds, γs)). sγs) I the preset pper we cosider the sme type of domi G s i [5] d give turl extesio of their result for the geerl model.). We lso preset some simultio results to illustrte the theoreticl oes where the Wieer sheet is simulted with the help of its Krhue-Loève expsio see e.g. [8]). 2. Model d estimtor Cosider the model.) with some give fuctios g,..., g p : R 2 R d with ukow regressio prmeters m,..., m p R. Let [, c] 0, ) d b, b 2, c), let γ,2 : [, b ] R d γ 0 : [b 2, c] R be cotiuous, strictly decresig fuctios d let γ : [b, c] R d γ 2 : [, b 2 ] R be cotiuous, strictly icresig fuctios with γ,2 b ) = γ b ) > 0, γ 2 b 2 ) = γ 0 b 2 ), γ,2 ) = γ 2 ) d γ c) = γ 0 c). Cosider the curve Γ := Γ,2 Γ Γ 2 Γ 0, where { s, Γ,2 := γ,2 s) ) } { s, : s [, b ], Γ := γ s) ) } : s [b, c], { s, Γ 2 := γ2 s) ) } { s, : s [, b 2 ], Γ 0 := γ0 s) ) } : s [b 2, c],

4 6 S. Br, K. Sikoly d for give ε > 0 let Γ ε,2, Γ ε, Γ ε 2 d Γ ε 0 deote the ier ε-strip of Γ,2, Γ, Γ 2 d Γ 0, respectively, tht is e.g. Γ ε,2 := { s, t) R 2 : s [, ε], t [γ,2 s), γ,2 )] or Suppose tht there exists ε > 0 such tht d cosider the set G := G G 2 G 3, where s [ ε, b ], t [γ,2 s), γ,2 s) ε] }. Γ ε Γ ε 2 = d Γ ε,2 Γ ε 0 =, 2.) G := { s, t) R 2 : s [, b b 2 ], t [γ,2 s), γ 2 s)] }, {{ s, t) R 2 : s [b, b 2 ], t [γ s), γ 2 s)] }, if b b 2, G 2 := { s, t) R 2 : s [b 2, b ], t [γ,2 s), γ 0 s)] }, if b > b 2, G 3 := { s, t) R 2 : s [b b 2, c], t [γ s), γ 0 s)] }. A exmple of such set of observtios c be see of Figure. t γ 0 γ 2 Γ 2 G Γ 0 Γ,2 Γ γ γ,2 b b 2 c s Figure : A exmple of set of observtios G The followig theorem is extesio of Theorem 2. of [5] d c be proved i similr wy. The proof is bsed o the discrete pproximtio method described i [3, 4, 5], which relies o the results of [, Sectio 2.3.2]. Theorem 2.. If g,..., g p re twice cotiuously differetible iside G d the prtil derivtives g i, 2 g i d 2 g i, i =,..., p, c be cotiuously exteded to G the the probbility mesures P Z d P W, geerted o CG) by the sheets Z d W, respectively, re equivlet d the Rdo-Nikodym derivtive of P Z with respect to P W equls { dp Z Z) = exp m dp W 2 Am 2ζ m )},

5 Prmeter estimtio i lier regressio drive by Wieer sheet 7 where A := A k,l ) p k,l=, m := m,..., m p ) d ζ := ζ,..., ζ p ) with A k,l := g k b, γ,2 b ) ) g l b, γ,2 b ) ) b γ,2 b ) [ gk s, γ,2 s) ) s g k s, γ,2 s) )][ g l s, γ,2 s) ) s g l s, γ,2 s) )] d b c b g k s, γ s) ) g l s, γ s) ) γ,2) γ,2b ) γ s) s 2 γ,2 s) ds 2 g k γ,2 t), t) 2 g l γ,2 t), t) γ 2b 2) γ 2) γ,2 t) dt ds 2.2) 2 g k γ 2 t), t) 2 g l γ 2 t), t) γ2 t) dt G 2 g k s, t) 2 g l s, t)ds dt, ζ k := g k b, γ,2 b ) ) Z b, γ,2 b ) ) c g k s, γ s) ) Z ds, γ s) ) 2.3) b γ,2 b ) γ s) b b [ gk s, γ,2 s) ) s g k s, γ,2 s) )] [Z s 2 s, γ,2 s) ) ds sz ds, γ,2 s) )] γ,2 s) γ 2b 2) γ 2) 2 g k γ 2 t), t) γ2 t) Z γ2 t), dt) 2 g k s, t) Zds, dt). G γ,2) γ,2b ) 2 g k γ,2 t), t) γ,2 t) Z γ,2 t), dt) If deta) 0 the the mximum likelihood estimtor of the prmeter vector m bsed o the observtios {Zs, t) : s, t) G} hs the form m = A ζ d hs p-dimesiol orml distributio with me m d covrice mtrix A. Remrk 2.2. Observe tht ll six terms of mtrix A re o-egtive defiite mtrices, so A is o-egtive defiite, too. Hece, to esure deta) 0 it suffices to hve t lest oe positive defiite mog the terms, which fulfils e.g. if g,..., g p re lierly idepedet. Remrk 2.3. We remrk tht the weighted L 2 -Riem itegrls of prtil derivtives of the Wieer sheet d of other L 2 -processes) log curve re defied i the sese of [5, Defiitio 4.]. This mes tht if Z is L 2 -process give log ε-eighborhood of curve Γ := { s, γs)) : s [, b] }, where γ : [, b] R is

6 8 S. Br, K. Sikoly strictly mootoe d y : [, b] R is fuctio, the b ys) Zds, γs)) := l.i.m. h 0 b ys) [ Zs h, γs)) Zs, γs)) ] ds, h γb) γb) yγ t)) Zγ t), dt) := l.i.m. yγ t)) [ Zγ t), t h) Zγ t), t) ] dt, h 0 h γ) if the right hd sides exist. Exmple 2.4. Cosider the model γ) Zs, t) = m s 2 t 2 ) m 2 s t) m 3 s t) W s, t), s, t) G, where W s, t), s, t) [u r, u r] [v r, v r] is stdrd Wieer sheet d G is circle with ceter t u, v) d rdius r. Thus γ,2 s) = v r 2 s u) 2, γ s) = v r 2 s u) 2, γ 2 s) = v r 2 s u) 2, γ 0 s) = v r 2 s u) 2, s [u r, u], s [u, u r], s [u r, u], s [u, u r], γ,2 t) = u r 2 t v) 2, t [v r, v], γ 2 t) = u r 2 t v) 2, t [v, v r]. I this cse the distict elemets of the symmetric mtrix A defied by 2.2) re the followig A, = u2 v r) 2 ) 2 uv r) v 4 v r t 2 γ,2 t)dt 4 u u r vr A,2 = u2 v r) 2 )uv r) uv r) A 2,2 = 2 ur u s γ s) ds 2 u v r)2 uv r) v v v r u u r γ,2s) 2 s 2 ) 2 s 2 ds 4 γ,2 s) t 2 γ 2 t)dt, u u r γ,2s) s 2 2 ds s 2 t γ,2 t)dt 2 vr v ur γ,2 s) s 2 ds u ur u t γ 2 t)dt, γ s) ds s 2 γ s) ds

7 Prmeter estimtio i lier regressio drive by Wieer sheet 9 v v r vr γ,2 t)dt A,3 = u 2 v r) 2 2ru2v)r 2, A 2,3 = uv2r, A 3,3 = uv r) rv2u) 3πr2 4, v γ2 2.4) t)dt, while the compoets of ζ = ζ, ζ 2, ζ 3 ) defied by 2.3) re ζ = ζ 2 = u 2 v r) 2) Z b, γ,2 b ) ) v v r u u r vr v 2t γ uv r) 2 t) Z γ2 t), dt) ur u 2s γ s) Z ds, γ s) ) γ,2s) 2 s 2 )[Z s, γ,2 s) ) ds sz ds, γ,2 s) )] s 2 γ,2 s) 2t γ,2 t) Z γ,2 t), dt), u v r ) Z b, γ,2 b ) ) v v r u u r uv r) ur u γ s) Z ds, γ s) ) γ2 t) Z γ2 t), dt) 2.5) [ s 2 Z s, γ,2 s) ) ds sz ds, γ,2 s) )] vr ζ 3 = Z b, γ,2 b ) ) vr v ur u Z γ,2 t), dt) Z ds, γ s) ) G Zds, dt). v v r v Z γ2 t), dt) γ,2 t) Z γ,2 t), dt),

8 0 S. Br, K. Sikoly 3. Simultio results To illustrte the theoreticl results of [2, 3, 4, 5] d of Theorem 2. we performed computer simultios usig Mtlb 200. I order to simulte Wieer sheet W s, t), 0 s S, 0 t T, we cosidered its Krhue-Loève expsio, tht is W s, t) j,k= ω j,k 8 ) ST π2j )t π 2 )2k )2j ) si si π2k )s ), 3.) where {ω j,k : j, k } re idepedet stdrd orml rdom vribles [8]. Figure 2 shows pproximtio of the Wieer sheet with = 50. Obviously, there re other methods of simultig Wieer sheet e.g. with the help of discretiztio d usig the idepedece of icremets see e.g. [5]). However, i order to clculte our estimtors we eed method which provides us whole reliztios of the sheet Figure 2: Simultio of Wieer sheet, = 50 I ech of the followig exmples 000 idepedet smples of the drivig Wieer sheet were simulted with vryig betwee 25 d 00 d the mes of the estimtes of the prmeters d the empiricl vrices or covrice mtrices of ζ defied by.3),.4) d 2.3), respectively, were clculted. Exmple 3.. Cosider the model Zs, t) = W s, t) ms 2 t 2 ),

9 Prmeter estimtio i lier regressio drive by Wieer sheet where W s, t), s, t) G = [, 3] 2 is stdrd Wieer sheet see Specil cse ). Compoets of A d of the pproximtio of ζ c be give i the followig closed form: A = 4b3 2 b b 2 b, ζ ma 8 { ST 2 b 2 ω j,k π 2 si )2k )2j ) b j,k= 2 ) [ π2j )b si cos b π2k ) ) π2k )2 2 si si [ ) π2j )b2 cos cos π2j ) ) π2j )b2 π2j )b b 2 si b si where ζ = 2 b 2 ) Z, b ) b π2k ) 2 ) ) π2k ) π2j )b si ) ) ) π2k )2 π2k ) cos ) ] 2 si ) ) π2j )b ) ]}, 2u b Z du, b ) b 2 b ) π2k ) 2v Z, dv ). The theoreticl prmeter vlue is m = 5, while A = O Figure 3 the mes of the estimtes of the prmeter d the estimted vrices of ζ re plotted versus the level of the pproximtio 3.). I cse of = 00 we hve m = d  = Exmple 3.2. Cosider the model Zs, t) = W s, t) m, where W s, t), s, t) G, is stdrd Wieer sheet d G is set stisfyig coditios of Specil cse 2 d Γ is prt of circle with ceter t the origi, tht is γs) = r 2 s 2 with some r > 0 d with [, b] 0, r) [4, Exmple.2]. The A = r 2 r2 2 b r2 ), c = 2 b r2 b 2 r 2, c 2 = r 2 r 2 b, 2 y s, r 2 s 2 ) 0, y 2 s, r 2 s 2 ) r 2

10 2 S. Br, K. Sikoly m  Figure 3: Mes of the estimtes of m d estimted vrices of ζ i Exmple 3. for m  Figure 4: Mes of the estimtes of m d estimted vrices of ζ i Exmple 3.2 for d ζ ma j,k= ω j,k 8 ST π 2 r 2 2k )2j )

11 Prmeter estimtio i lier regressio drive by Wieer sheet 3 { ) r2 2 π2k ) π2j ) ) r si si 2 2 ) b π2k )b r2 b si π2j ) ) r si 2 b 2 2 b { ) π2k )s π2k )s r 2 s cos π2j ) ) r si 2 s 2 2 π2j ) ) π2k )s π2j ) ) } r si cos 2 s ds} 2, where ζ is defied by.4). Let prmeter vlue be m = 5 d choose =, b = 3 d r = 5 yieldig A = O Figure 4 the mes of the estimtes of the prmeter d the estimted vrices of ζ re plotted versus the level of the pproximtio 3.). I cse of = 00 we hve m = d  = m m m Figure 5: Mes of the estimtes of the compoets of m i Exmple 3.3 for Exmple 3.3. Cosider the sme model Zs, t) = m s 2 t 2 ) m 2 s t) m 3 s t) W s, t), s, t) G, s i Exmple 2.4, where W s, t), s, t) [0, 8] 2, is stdrd Wieer sheet d G is circle with ceter t 6, 6) d rdius r = 2. I this cse the etries

12 4 S. Br, K. Sikoly of the mtrix A defied by 2.4) d the pproximtio of the compoets of ζ = ζ, ζ 2, ζ 3 ) defied by 2.5) c be clculted usig umericl itegrtio, where Mtlb fuctio qud is pplied recursive dptive Simpso qudrture) Â, 335 Â, Â2,2 5.9 Â, Â3,3 50 Â2, Figure 6: Estimted covrices of ζ i Exmple 3.3 for The theoreticl prmeter vlues re m = 5, m 2 = 8 d m 3 = 3, while the theoreticl covrice mtrix of ζ equls A = O Figure 5 the mes of the estimtes of the three prmeters, while o Figure 6 the estimted covrices of ζ re plotted versus the level of the pproximtio 3.). I cse of = 00 we hve , , ) for the me d  = for the covrice mtrix.

13 Prmeter estimtio i lier regressio drive by Wieer sheet 5 Refereces [] Artó, M. 982) Lier Stochstic Systems with Costt Coefficiets. A Sttisticl Approch. Spriger-Verlg, Berli; i Russi: Nuk, Moscow, 989. [2] Artó, N. M. 997) Me estimtio of Browi sheet. Comput. Mth. Appl. 33, o. 8, [3] Br, S., Pp, G. d Zuijle, M. v. 2003) Estimtio of the me of sttiory d osttiory Orstei-Uhlebeck processes d sheets. Comput. Mth. Appl. 45, o. 4-5, [4] Br, S., Pp, G. d Zuijle, M. v. 2004) Estimtio of the me of Wieer sheet. Stt. Iferece Stoch. Process. 7, o. 3, [5] Br, S., Pp, G., Zuijle M. v. 20) Prmeter estimtio of shifted Wieer sheet. Sttistics 45, o. 4, [6] Biffis, E. d Millossovich, P. 2006) A bidimesiol pproch to mortlity risk. Decisios Eco. Fi. 29, o. 2, [7] Crter, A. V. 2006) A cotiuous Gussi pproximtio to oprmetric regressio i two dimesios. Beroulli 2, o., [8] Deheuvels, P., Peccti, G. d Yor, M. 2006) O qudrtic fuctiols of the Browi sheet d relted processes. Stochstic Process. Appl. 6, o. 3, [9] Dougls, J. F. 996) Swellig d growth of polymers, membres, d spoges. Phys. Rev. E 54, o. 3, [0] Goldstei, R. 2000) The term structure of iterest rtes s rdom field. Rev. Fic. Stud. 3, o. 2, [] Kuo, H. H. 975) Gussi Mesures i Bch Spces. Spriger-Verlg, Berli. [2] Kurod, K. d Mk, H. 987) The iterfce of the Isig model d the Browi sheet. J. Stt. Phys. 47, o. 5-6, [3] Rozov, Yu. A. 968) Gussi Ifiite-dimesiol Distributios. i Russi). Tr. Mt. Ist. Akd. Nuk SSSR, Nuk, Moscow. [4] Rozov, Yu. A. 990) Some boudry problems for geerlized rdom fields. Theor. Probb. Appl. 35, o. 4, [5] Toth, D. 2004) Addig iterior poits to existig Browi sheet lttice. Sttist. Probb. Lett. 66, o. 3,